PhDThesis_Hartkamp

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ISBN 978-90-365-3532-8

AM OL E C UL ARDY NAMI C S S T UDY OF N O N- NE WT O NI A NF L O W S OF S I MP L E F L UI D S I N C O NF I NE DA ND U N C O NF I NE D GE OME T RI E S

R e m c oH a r t k a m p

A MOLECULAR DYNAMICS STUDY OFNON-NEWTONIAN FLOWS OF SIMPLE FLUIDS IN

CONFINED AND UNCONFINED GEOMETRIES

Remco Hartkamp

Invitation

To the public defenseof my thesis

A MOLECULARDYNAMICS STUDY

OFNON-NEWTONIANFLOWS OF SIMPLE

FLUIDS IN CONFINEDAND UNCONFINED

GEOMETRIES

onWednesday the 15 th of May 2013

at 14:45in the prof. dr. G. Berkhoff-room

of the Waaier building of theUniversity of Twente in Enschede

Before the defense, at 14:30, I willgive a brief introductory talk on the

topic of my thesis.

After the defense there will be areception.

Remco [email protected]

Tel: 06-30057263


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A MOLECULAR DYNAMICS STUDY OF

NON-NEWTONIAN FLOWS OF SIMPLE FLUIDS INCONFINED AND UNCONFINED GEOMETRIES

Remco Hartkamp


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This dissertation has been approved by:

prof. dr. S. Luding

prof. dr. B. D. Todd

Thesis committee members:

prof. dr. F. Eising University of Twente, chairman/secretaryprof. dr. S. Luding University of Twente, supervisorprof. dr. B. D. Todd Swinburne University of Technology, supervisorprof. dr. A. van den Berg University of Twenteprof. dr. W. J. Briels University of Twenteprof. dr. D. M. Heyes Imperial College Londonprof. dr. J. Westerweel Delft University of Technologydr. J. S. Hansen Roskilde Universitydr. T. Weinhart University of Twente

A molecular dynamics study of non-Newtonian ows of simple uidsin conned and unconned geometriesRemco Hartkamp

Printed by Gildeprint DrukkerijenThesis University of Twente, Enschede, The NetherlandsISBN 978-90-365-3532-8

Copyright c 2013 by Remco Hartkamp, The Netherlands


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A MOLECULAR DYNAMICS STUDY OF

NON-NEWTONIAN FLOWS OF SIMPLE FLUIDS IN

CONFINED AND UNCONFINED GEOMETRIES

DISSERTATION

to obtainthe degree of doctor at the University of Twente,

on the authority of the rector magnicus,prof. dr. H. Brinksma,

on account of the decision of the graduation committee,to be publicly defended

on Wednesday the 15 th of May 2013 at 14:45

by

Remco Marcel Hartkamp

born on 31 July 1985

in Amsterdam, The Netherlands


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Summary

Various uid ow phenomena originate in the dynamics of the atoms that constitutethe uid. Studying uids as a collection of atoms is key to a better understanding of,for example, non-Newtonian uid ow behavior. Molecular dynamics (MD) is a verysuitable tool for the study of uids on the atomic level. Many MD studies have beendevoted to the behavior of homogeneous, unconned uids under either simple shearor extensional ows, while a combination of both ow types has not been studiedextensively. Strongly conned, inhomogeneous uids are usually studied separatelyfrom homogeneous uid problems because of their very different behavior, due to wall-

effects. In this thesis, a unied approach is developed, to study and compare thestresses in different ow situations.

We use MD simulations and analysis tools for: (1) the study of various properties of a simple homogeneous bulk uid under several planar velocity elds, (2) the calculationof stresses and viscosity using the transient-time correlation function and, (3) the studyof properties of an inhomogeneous uid conned in a nanochannel.

The data suggest that the pressure tensor for a homogeneous, simple, monoatomicuid under any planar ow eld can be expressed in a unied form as a combinationof equilibrium properties and non-Newtonian phenomena, such as: strain thinning vis-cosity, viscoelastic lagging, pressure dilatancy and out-of-ow plane anisotropy. Wefound consistent trends for these non-Newtonian quantities as a function of the mag-nitude of the strain rate tensor and the vorticity, at different state points. Similarly,interesting trends have been found for equilibrium material properties, such as thezero-shear rate rst normal stress coefficient, as a function of density.

It is often not possible to directly compare experimental data to results from steadynon-equilibrium molecular dynamics (NEMD) simulations. Calculating accurate time-averaged values from these simulations is usually only feasible at deformation rates

that are much larger than those accessible in experiments. We have shown that thetransient-time correlation function provides a more efficient alternative to direct time-

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averaging of NEMD data. This method has been applied to an atomic uid undercombined shear and planar elongational ow, and to molecular uids under varioustypes of planar ow.

Non-Newtonian stresses have been studied for a simple monoatomic uid connedin a nanochannel, where the properties vary across the channel. The pressure tensorhas been expressed in terms of objective quantities, as a function of the position acrossthe channel due to layering of the atoms. Data for various densities, temperatures andbody forces have provided insight in the dependencies of various quantities. Relatingthe objective quantities derived from the stress tensor to local values of other statevariables has not yet been fully achieved and a purely local relation between them maynot exist, leaving many open questions for future research.

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Samenvatting

Velerlei stromingsfenomenen hebben een oorsprong in de bewegingen van de atomenwaaruit een vloeistof bestaat. Het bestuderen van een vloeistof als een verzamel-ing atomen is een belangrijk component voor het verkrijgen van een beter begrip bijvoorbeeld van niet-Newtoniaans stromingsgedrag. Moleculaire dynamica (MD) is eenerg geschikt hulpmiddel voor het bestuderen van vloeistoffen op een atomair niveau.Vele MD studies zijn gewijd aan homogene onbegrensde vloeistoffen onder invloedvan afschuiving of extensie stromingen. Sterk begrensde, inhomogene vloeistoffen zijnmeestal afzonderlijk van homogene vloeistoffen bestudeerd vanwege hun sterk verschil-

lende gedrag, veroorzaakt door wand-effecten. Een aanpak is ontwikkeld in deze scrip-tie om de spanningen in een vloeistof onder verschillende stromingen op een uniformemanier te bestuderen en te vergelijken.

We gebruiken MD simulaties en analyse hulpmiddelen voor: (1) het bestuderen vanallerlei eigenschappen van simpele homogene vloeistoffen onder invloed van verschil-lende planaire snelheidsvelden, (2) het berekenen van spanningen en viscositeit doormiddel van de transient-tijd correlatie functie en, (3) het bestuderen van eigenschappenvan een inhomogene vloeistof in een nanokanaal.

De data suggereert dat de spanningstensor voor een homogene, simpele, edelevloeistof onder ieder planair snelheidsveld kan uitgedrukt worden in een uniformevorm in termen van evenwichtsgrootheden en niet-Newtoniaanse fenomenen, zoals:afnemende viscositeit onder deformatie, viscoelastische vertraging, verhoging van dedruk onder deformatie en anisotropie in de richting haaks op het stromingsveld. Wehebben consistente trends gevonden voor deze grootheden als functie van de van desterkte van de schuifsnelheidstensor en de vorticiteit voor vloeistoffen met verschil-lende dichtheden en temperaturen. Verder hebben we interessante trends gevondenvoor de materiaaleigenschappen van een vloeistof in evenwicht, bijvoorbeeld de nul-

afschuivingssnelheid eerste normaalspanningscoefficient als een functie van de dichtheid.Het is vaak niet mogelijk om experimentele data direct te vergelijken met resul-

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taten van stationaire moleculaire dynamica simulaties uit evenwicht. Het nauwkeurigberekenen van tijd-gemiddelde waarden met behulp van dit soort simulaties is over hetalgemeen alleen mogelijk als de afschuifsnelheid veel groter zijn dan wat haalbaar isin experimenten. We hebben laten zien dat de transient-tijd correlatie functie een ef-cienter alternatief biedt dan het direct middelen van moleculaire dynamica simulatiedata. Deze methode is toegepast op een atomaire vloeistof onder een combinatie vanafschuiving en planaire extensie stroming, en op moleculaire vloeistoffen onder allerleitypes of planaire stroming.

Niet-Newtoniaanse spanningen zijn bestudeerd voor een simpele atomaire vloeistof in een nanokanaal, waar de vloeistofeigenschappen varieren met de positie in hetkanaal. De spanningstensor is uitgedrukt in termen van objectieve grootheden, als

functie van de positie in het kanaal als gevolg van laagvorming van de atomen. Datavoor verschillende vloeistofdichtheden, temperaturen en drijfkrachten hebben inzichtverschaft in de samenhang tussen velerlei grootheden. Het relateren van de objectievegrootheden die afgeleid zijn van de spanningstensor aan andere grootheden is nog nietvolledig bereikt en het bestaan van een volledig lokale relatie is onzeker, dit is een openprobleem voor toekomstig onderzoek.

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Contents

Summary i

Samenvatting iii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Historical notes and literature . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Hydrodynamics and transport coefficients . . . . . . . . . . . . . . . . . 51.4 The study of uids with MD . . . . . . . . . . . . . . . . . . . . . . . . 71.5 MD simulations of very simple systems . . . . . . . . . . . . . . . . . . . 101.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Molecular Dynamics Simulations 15

2.1 Integration schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.1 (Velocity) Verlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Runge-Kutta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Non-bonded interactions between atoms . . . . . . . . . . . . . . . . . . 202.3 Lennard-Jones Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Pressure and stress tensors . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Thermostatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 Gaussian thermostat . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.2 Nose-Hoover thermostat . . . . . . . . . . . . . . . . . . . . . . . 362.5.3 Braga-Travis congurational thermostat . . . . . . . . . . . . . . 38

2.6 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 392.7 Non-equilibrium molecular dynamics . . . . . . . . . . . . . . . . . . . . 43

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CONTENTS

3 Homogeneous non-equilibrium molecular dynamics simulations 47

3.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.1 Thermostatted SLLOD . . . . . . . . . . . . . . . . . . . . . . . 513.1.2 Molecular SLLOD . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Shear ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Elongational ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1 Planar Elongational Flow . . . . . . . . . . . . . . . . . . . . . . 563.3.2 Kraynik-Reinelt Periodic boundary conditions . . . . . . . . . . . 58

3.4 Combined shear and elongational ows . . . . . . . . . . . . . . . . . . . 61

4 Statistical mechanics 69

4.1 General phase space description . . . . . . . . . . . . . . . . . . . . . . . 694.2 Statistical ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Time-correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4 Calculation of Navier-Stokes transport coefficients . . . . . . . . . . . . 774.5 Transient-time correlation function . . . . . . . . . . . . . . . . . . . . . 794.6 Pair distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Non-Newtonian pressure tensors for simple atomic uids 85

5.1 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1.1 Introduction to viscoelasticity: Cyclic deformation . . . . . . . . 875.1.2 Material functions for viscoelastic uids . . . . . . . . . . . . . . 89

5.2 Calculation of material constants from EMD simulations . . . . . . . . . 935.2.1 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Non-Newtonian constitutive models . . . . . . . . . . . . . . . . . . . . 1035.3.1 Rotating the pressure tensor . . . . . . . . . . . . . . . . . . . . 1125.3.2 An objective model to describe and predict the non-Newtonian

pressure tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4 Transient ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.4.1 Startup ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.4.2 Relaxation to equilibrium . . . . . . . . . . . . . . . . . . . . . . 122

5.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Transient-time correlation functions applied to atomic and molecularuids 131

6.1 Transient-time correlation function . . . . . . . . . . . . . . . . . . . . . 133

6.2 Planar Mixed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2.1 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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CONTENTS

6.2.2 Atomic mixed ow results . . . . . . . . . . . . . . . . . . . . . . 1396.3 Normal stress differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.4 Molecular TTCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4.1 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7 Conned atomic uids 1637.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.2 Simulating atomistic channels . . . . . . . . . . . . . . . . . . . . . . . . 1707.3 Spatial averaging and macroscopic elds . . . . . . . . . . . . . . . . . . 174

7.3.1 Streaming velocity, strain rate and temperature . . . . . . . . . . 1787.3.2 Stress calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.4 A study of a uid conned in a nanochannel . . . . . . . . . . . . . . . . 1817.4.1 Model system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.4.2 Constitutive model with anisotropic stress . . . . . . . . . . . . . 1847.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.4.4 Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . 206

7.5 Conclusions and summary . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8 Conclusions and recommendations 217

8.1 Homogeneous uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2178.2 Conned uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

A Derivation of the SLLOD equations of motion 223

B Validation of our objective model for the pressure tensor 227

Bibliography 228

Acknowledgement 261

Curriculum vitae 263

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1

Introduction

1.1 Motivation

In 1959, Richard Feynman gave a lecture that was titled: Theres plenty of room at the bottom [1]. This lecture is widely seen as one of the main inspirations that has led tothe rapidly developing eld of nanotechnology. 1 The eld is growing at an incrediblerate and more than 30% of all scientic publications in the European Union wererelated to nanotechnology in 2006 [3]. Besides its signicant role in the scientic world,nanotechnology has also been a source of inspiration for various science-ction writersas well as makers of games and movies [4]. Many scientic and non-scientic authorshave speculated in the past decades about possible applications of nanotechnologyand their impact on the society and economy worldwide [ 5, 6]. Many of the predicteddevices have not materialized yet, but the eld has come a long way in the pastdecades with the development, fabrication and miniaturization of microdevices andnanodevices [711] that contain very small channels, bearings, valves or nozzles. Theeld is involved with mechanical, electrical, chemical and rheological processes. Thisthesis will be devoted to the study of uids on a molecular level.

Our understanding of physics on the molecular level has lagged behind develop-ments in the fabrication of new, smaller or improved devices. Theoretical, computa-tional and experimental studies can lead to an increased understanding of phenomenaand their origins. While some experiments [ 1216] could predict effective global prop-erties like relaxation time, frictional force or shear response of ultra-thin liquid lms,the extraction of local values of state variables (like density, pressure and temperature)is still beyond the reach of experimental measurements. On the other hand, such localquantities can be extracted rather easily from computer simulations. Computationalstudies of uids can either be done by assuming that the uid can be approximated

1 The term nanotechnology was rst used by Tanuguchi [2], in 1974.

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1.1. MOTIVATION

as a continuous medium, or from a rst-principles approach, that treats the uids asa collection of molecules. Both approaches are discussed in this chapter.

In the former approach, the system is subdivided in small uid elements. The sizeof a uid element has to be big enough so that it corresponds to a sufficiently largenumber of molecules, such that random uctuations of molecules have no notable effecton instantaneous local quantities. On the other hand, solving a ow problem usingvery large uid elements results in a poor spatial resolution. Many ow problemsinvolve a ow through or around an object. When this is the case, suitable boundaryconditions are needed in order to calculate the macroscopic quantities (e.g., density,pressure, velocity and temperature) in each uid element. Many continuum modelsassume a no-slip boundary condition for the uid-solid interface (i.e., the uid velocity

equals the velocity of the interface at the location of contact). Furthermore, continuummodels require prior knowledge about the transport properties of the uid in order tocalculate the macroscopic quantities in each uid element.

The use of uid elements that are much larger than the molecular diameters isoften not permitted, or even possible, in geometries that have a characteristic lengthin the micrometer or nanometer range .2 Also, the no-slip boundary condition is knownto be inaccurate on the molecular scale.

A rst-principles approach, such as molecular dynamics (MD, see Chapter 2), doesnot suffer from this limitation. How much mass, momentum and energy are trans-

ported, or how large the slip length of a uid-solid interface is, are results of MDsimulations, rather than being required prior knowledge to address a ow problem.Therefore, MD is a suitable method for the study of dense conned owing liquids.

The behavior of uids in a nano-conned geometry often deviates from those inlarger geometries. This is due to a lack of separation between the characteristic lengthscale of the system and the atomic length scale. For example, the balance betweensurface forces (such as pressure) and volume forces (such as gravity) shifts towards thesurface forces as the length scale of the problem decreases. A good understanding of dynamical, structural and chemical properties of materials on a molecular level benetsthe development and improvement of nanodevices and opens the door to possible newapplications.

For the study of a bulk uid, the continuum approximation may be permittedand wall slip is not considered in that case. However, a continuum treatment of theproblem might still not be preferred over a rst-principles simulation method. A con-tinuum treatment with phenomenological closure relations would require knowledgefrom experimental studies, which is not always available and much more costly thancomputer simulations. Deviation from Newtonian rheology nds its origin in the mi-

2

Whether this assumption holds depends not only on the size of the system, but also on the typeof uid and on its density and temperature [ 11].

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1.2. HISTORICAL NOTES AND LITERATURE

crostructure of a uid. Since nding a molecule-based theory to describe the rheologyof dense liquids is still an open problem [10], a rst-principles computational study isagain suitable to increase the understanding of non-Newtonian rheology of uids thatbehave Newtonian under certain conditions. The literature about non-Newtonian rhe-ology is vast and mainly concentrated on polymeric materials. Yet, even simple atomicuids exhibit non-Newtonian phenomena under large enough deformation rates. Theseuids are computationally cheaper to study and their simpler dynamics can show moreclearly what mechanisms are responsible for non-Newtonian phenomena. The samemechanisms may play an important role in the rheology of polymers or other complexuids.

MD simulations are highly computationally expensive compared to a continuum

approach. This is due to the fact that the interactions of atoms with many surround-ing atoms need to be calculated at every time step and that the spatial resolution inMD is much larger than in continuum methods, where a uid element needs to belarge compared to atoms. The computational cost poses a limitation on the num-ber of atoms and the number of time steps in a typical MD simulation. Dependingon the given uid problem, other particle-based methods could be more suitable orcomputationally cheaper. Examples are: Monte Carlo (MC), direct simulation MonteCarlo (DSMC), stochastic rotational dynamics (SRD), Brownian dynamics (BD), den-sity functional theory (DFT), smoothed-particle hydrodynamics (SPH) and dissipative

particle dynamics (DPD). Many of these methods are, compared to MD, less suitablefor the simulation of dense liquids and they will not be discussed in this work. For adiscussion of the application of various of these methods to molecular modeling, seefor example Ref. [17].

In this thesis, we use MD simulations in conjunction with statistical mechanicsmethods to study the structural and dynamical properties of simple atomic uids,such as Argon, Krypton or Xenon ,3 in conned and unconned geometries. Much of the work presented in this thesis is related to calculating stresses and shear viscosity,quantifying non-Newtonian stress behavior and developing a unied approach to studythe stresses in a uid and express them in quantities that are invariant to rotation andtranslation of the coordinate system.

1.2 Historical notes and literature

Thermodynamics and statistical mechanics form the basis of many methods that areused in molecular dynamics. While thermodynamics and statistical mechanics go far

3

These uids exhibit no chemical reactions and are mono-atomic noble gases that can be approx-imated well by spheres.

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1.2. HISTORICAL NOTES AND LITERATURE

back, computer simulations came relatively recently, rst in the form of equilibriummolecular dynamics (EMD) simulations. In 1959, Alder and Wainwright [ 18] were therst to report results from simulations that contained more than 100 interacting hard-sphere particles. They showed trajectories of particles in a two-dimensional liquidand vapor in a periodic cell. The authors realized that this new simulation methodoffered a tool to solve many open problems in statistical mechanics. The problemthey addressed in particular still raises questions nowadays: could Newtons equationsof motion, which are reversible, account for the irreversible thermodynamics, suchas an increasing entropy? A few years later, in 1964, Rahman[ 19] used a Lennard-Jones potential to simulate a liquid consisting of 864 particles. He calculated variousproperties, such as the velocity autocorrelation function, the mean-square displacement

and the pair distribution function. While computers have become immensely morepowerful since the 60s, the methods that were used in these two studies are stillcommonly used in modern studies.

In 1975, a much more efficient and direct approach to calculate transport coef-cients was developed by Hoover and Ashurst [ 20], which is called non-equilibriummolecular dynamics (NEMD). In this method, the uid is driven away from thermo-dynamic equilibrium by a thermal or mechanical external eld or by solid bound-aries, much alike actual experiments. The same authors later used NEMD to look athard-sphere models and Lennard-Jones uids in order to compare their results to the

Green-Kubo transport coefficients [ 21, 22]. They found good agreement between theirresults and the shear viscosity of a Lennard-Jones uid computed by Levesque et al.[23], who used EMD. Furthermore, the simulations of Hoover and Ashurst showed,for a soft-sphere uid, the same deviation from Enskog theory that was found earlierby Alder et al. [24] for a hard-sphere uid. The development of NEMD gave riseto a whole range of new simulations and has developed, over the last decades, into alarge eld of study that has provided insight in the micro-structural origins of variousmacroscopic phenomena.

Hoover et al. [25] showed that the DOLLS 4 Hamiltonian, that couples a homoge-neous driving eld to the uid, could be used to simulate adiabatic ows. The equationsof motion that can be derived from this Hamiltonian are called the DOLLS equations of motion. Shortly after, Ladd [ 26] found that the DOLLS shear ow algorithm producedincorrect normal stress differences. Investigators, such as Ladd, Hoover, Evans andMorriss then developed the SLLOD equations of motion [2628]. This homogeneous-ow algorithm has later, in conjunction with suitable periodic boundary conditions,also been used to simulate planar elongational ow [2931] and combined shear andelongational ow [32].

4

Named after the Kewpie doll, where Kewpie is replaced by q- p, that represents the particlespositions and peculiar momenta, respectively.

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1.3. HYDRODYNAMICS AND TRANSPORT COEFFICIENTS

Magda et al. [33] and Bitsanis et al. [3436] were among the rst to study stronglyconned atomic liquids using MD simulations. They conned a Lennard-Jones uidbetween two solid walls where the functional form of the uid-wall interaction potential,as a function of the distance from the wall, accounted for the structure of the walls.They found a density prole that varied with the position across the channel.

The application of NEMD to conned uids came shortly after: Liem et al. [37]mimicked a real shear-ow experiment by conning and driving the uid via slidingatomistic walls. Harmonic springs were used to connect the particles to their respectivesites in a hexagonal lattice. This approach is still used nowadays in many conned-uid simulations. The authors compared the results from the boundary-driven shearsimulations to homogeneous shear simulations in order to validate the correctness of

the homogeneous shear approach. The authors argued that the homogeneous-shearalgorithm removes the excess heat in an unphysical way by thermostatting the uideverywhere in the system, whereas heat is removed via the walls in conned-uidsimulations, as is the case in experiments. It was found that the pressure tensorsobtained from both approaches were in good agreement when the uid is sheared atsmall shear rates. When the uid is sheared too fast, viscous heat is generated fasterthan it is transported and the two approaches lead to different results.

1.3 Hydrodynamics and transport coefficientsContinuum methods deal with small volume elements that contain a number of atomsof the order of Avogadros constant N A = 6 .02 1023 . Macroscopic quantities in auid (e.g., density, pressure, temperature) can vary with position and time, but areassumed to be homogeneous over macroscopically small uid elements.

The governing set of equations in many continuum methods for uids are basedon the evolution of conserved quantities, for example: mass, linear momentum andenergy, and their rst gradients. The evolution of these quantities are respectively

described byt

= J , (1.1) Jt

= (uu + P ) + F E , (1.2)et

= (eu + J Q ) P T :u . (1.3)The mass density is denoted by , u is the streaming velocity vector, uu is the dyadicproduct, J = u is the momentum density, P is the second-order pressure tensor, P T

is its transpose, FE

is an external force per unit volume, e is the internal energy perunit volume and J Q is the heat ux vector.

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1.3. HYDRODYNAMICS AND TRANSPORT COEFFICIENTS

This set of equations is not closed, i.e., it cannot be solved uniquely yet becausethe number of unknowns is larger than the number of equations. Thus, additionalrelations are needed to close the system. For example, a set of constitutive equationscan be introduced, that relate forces and uxes by means of transport coefficients.These coefficients can be determined either experimentally or from MD simulationdata using statistical mechanics principles. In special cases, kinetic theory providesanalytical predictions. Once the transport coefficients are known, a continuum uidsolver can be used to solve the closed system of governing equations.

The validity of a constitutive relation can be limited to certain conditions. Forexample, a relation might break down if the system is far from thermodynamic equi-librium, e.g., if the shear rate becomes too large. The critical value of the shear rate

depends on the uid and on its state point. A uid is considered to be close to equi-librium if the following two postulates hold [28]: Firstly, the local thermodynamicequilibrium hypothesis needs to be satised. This hypothesis states that the principlesof equilibrium thermodynamics hold for uids close to equilibrium. In practice, thismeans that the gradients of macroscopic elds in the volume element are negligiblysmall if the driving force is small enough. The system is then said to be globally closeto equilibrium and locally in equilibrium. Secondly, the entropy source strength s forsystems close to equilibrium takes the canonical form

s =i

J i

X i , (1.4)

where J i are the phenomenological uxes that are associated with irreversible phe-nomena and X i are the conjugate thermodynamic forces. If these postulates hold, itis possible to relate the thermodynamic forces that occur in the entropy production toconjugate thermodynamic uxes via linear transport coefficients L ij

J i =j

L ij X j . (1.5)

This constitutive relation shows that when a thermodynamic force vanishes, then so

will its corresponding ux and its contribution to entropy production. Substitutingthe appropriate constitutive relations into the governing balance equations (Eqs. ( 1.1),(1.2) and ( 1.3)) reduces the number of unknowns, making it possible to solve the setof equations. This is often so complicated that solving it analytically is not feasible,such that a numerical solver is needed. The transport coefficient for a uid is denedas a rank two tensor in general, but can be reduced to a scalar quantity in all casesconsidered in this study. In conventional continuum hydrodynamics, the followinglinear constitutive relations are often used to close the system

J Q = T Fourier

s law of heat conductionP = pI (u + (u )T ) v 23 ( u )I Newton s law of viscosity

(1.6)

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1.4. THE STUDY OF FLUIDS WITH MD

where is the thermal conductivity, T is the temperature, p is the hydrostatic pressureand I the identity tensor, is the shear viscosity of the uid and v is its bulk viscosity.These closure relations are used in conjunction with assumptions, such as laminarow and symmetry of the pressure tensor and an equation of state to calculate thehydrostatic pressure p and a relation to relate the temperature to the internal energy.

1.4 The study of uids with MD

The MD approach is microscopic, meaning that it calculates the trajectories of discreteobjects in a many-body system. This approach is unfeasible for systems that containa number of particles of the order of Avogadros number, due to the enormous amountof data and computational time required. The number of particles in a MD simulationcan range from hundreds to millions. A characteristic time scale in MD is given by thecollision time, which is typically of the order of 10 15 s. Hydrodynamic time scales,on the other hand, are coupled to the speed of sound and are typically of the orderof seconds, or even larger. Dimensionless quantities, such as the Reynolds numberRe = uL , can be used as a characteristic number to classify ows on different scalesaccording to the ratio of inertial and viscous forces. The Reynolds numbers in MDsimulations are relatively low ( O(1)) in most cases, but turbulence can be observed insome MD simulations [3840]. As opposed to continuum solvers, MD simulations donot require specic models when turbulence is present in the uid.

MD simulations can be used for the study of uid problems where continuummethods are not suitable, for example strongly inhomogeneous uids. However, thegap between the characteristic length and time scales in both methods need to bebridged in order to compare the MD results to continuum theory or couple MD resultsto a continuum solver, as is done in multiscale methods [41, 42].

Many ows in nature and industry can be studied using (a combination of) linearvelocity proles. Figure 1.1 shows some common (simplied) types of ow for a bulkuid (i.e., far from any solid interface). The square represents a uid element and thearrows outside of the cell indicate the deformations applied to the uid. The arrowsparallel to a surface represent a shear deformation, whereas arrows perpendicular to asurface represent an expansion or contraction of the uid.

The most widely studied type of ow is (simple) shear ow. MD simulations of shear ow can either make use of homogeneous shear algorithms [4346] or boundary-driven simulations [ 37, 4749]. The former type of simulation models a bulk uid,whereas the latter attempts to mimic a conned-uid experiment by including walls

in the simulation. Both simulation types are a subset of NEMD. What differentiatesNEMD from EMD is the presence of an external force that drives the system away from

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x

y

Shear Elongation Mixed

Figure 1.1: A schematic of three planar ow types.

thermodynamic equilibrium and leads to a non-zero net transport of mass, momentumor energy. NEMD has seen a huge growth in recent years because of its potential tostudy transport coefficients and rheological properties of uids in an efficient way bymimicking real experiments.

Both types of NEMD simulations are based on the same principles: Solving thegoverning equations of motion that describe the evolution of the phase space variables:the positions and velocities of atoms. Non-equilibrium statistical mechanics methodsare used to calculate various physical quantities from the phase space variables. Thisis often far more difficult than in equilibrium, since some processes and denitions are

less well-understood or not well-dened out of equilibrium. Temperature is an exampleof a quantity that is not uniquely dened out of equilibrium. For example, Hoover andHoover [50] compared three non-equilibrium denitions of temperature.

Simulations of unconned uids can be used to study the properties of a bulk uid,i.e., a uid that is far enough removed from a conning surface, such that the surfacedoes not affect the uid. The boundaries of the simulation cell in such simulations aregenerally periodic and may not affect or disturb the ow in any way. Bulk uids undera homogeneous ow eld can be simulated by integrating a suitable set of equations of motion that couple a velocity gradient homogeneously to the atoms, i.e., the absolutepositions of the atoms have no explicit relevance, only their positions relative to eachother do. Since the uids in this type of simulations are homogeneous, informationcan be averaged over space and the number of particles required for such simulationsis typically small ( N = O(103)).

Conned uid simulations often aim to mimic an experiment or real-life problemin a natural way. Walls, a free surface or heat sources and sinks are explicitly modeledto replicate a real system. This type of simulation typically requires a much largernumber of particle ( N = O(104 107)) than the homogeneous approach, in which uidproperties are independent of the position. The presence of a solid surface causes thedistribution of particle positions (and thus also the density of the uid) to become a

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x

y

f

Figure 1.2: A schematic of planar Poiseuille ow.

function of the position, which affects other quantities as well. Variations in density

damp out over a distance of several atomic length scales and these variations very closeto an interface can often be considered negligible in large enough systems. Whenthe size of the system is within an order of magnitude from the atomic length scale,variations cannot be ignored and averaging over the volume is no longer permitted.The behavior of strongly conned, inhomogeneous uids is far from understood. Inrecent years, many efforts have been made to study the transport coefficients of uidsin conned geometries and to nd a constitutive relation that couples the microscopicdata to macroscopic balance equations. Finding such suitable constitutive relationsremains an open problem for strongly conned uids.

Solid boundaries need to be explicitly included in MD simulations in some cases.An example is (planar) Poiseuille ow, shown in Figure 1.2. In this ow type, a uidis conned between two parallel solid interfaces and is driven by a pressure differenceor a body force, for example gravity. As the body force pushes the uid to accelerate,viscous effects are responsible for a resistance to ow. A steady ow can form when thedriving force and the viscous resistance are in balance. The resulting velocity proleis typically a quadratic function of the position across the channel.

Even simple atomic uids are known to exhibit non-Newtonian phenomena, suchas shear thinning, shear dilatancy and normal stress effects [51, 52]. Furthermore,the density in strongly conned uids can become inhomogeneous, which affects thetransport properties as well as the phase diagram (i.e., a diagram that shows in whichphase or phases a material can occur at a given state point) and complicates the searchfor a constitutive law that describes the rheology of conned uids. Material constants,like shear viscosity, bulk viscosity, rst and second normal stress coefficients, shearrelaxation modulus and bulk relaxation modulus are dened to describe Newtonianand non-Newtonian properties. With the present work we aim to make contributions

to the elds of homogeneous MD simulations as well as increasing understanding of the properties of highly conned inhomogeneous uids.

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1.5. MD SIMULATIONS OF VERY SIMPLE SYSTEMS

1.5 MD simulations of very simple systems

Simulating large and complicated systems was not feasible in the early days of MD

simulations. Computers have become more powerful in the past decades and moremethods for simulating and analyzing complicated molecular systems have been de-veloped. While very complicated systems can now be studied with MD, there still isa scientic interest in simulations of simple uids in a periodic simulation cell or ina conned geometry. The main goal of these simulations is often not an attempt torecreate experimental systems as closely as possible, while some MD simulations of more complicated materials may have this purpose [5355].

Many experiments and devices that have been engineered in the past decade operateon length scales of at least several hundreds of nanometers wide, and more often in themicrometer range. 5 For the development of smaller devices one has to face challenges inthe methods of fabrication, detection, ow control and surface modication [56]. ManyMD simulations of simple conned uids, on the other hand, have a characteristiclength scale around 1 10 nm. Simulations of larger systems contain many moreatoms and thus become very computationally expensive. Furthermore, there is ascientic interest in understanding the dynamics and structural properties of uidsconned in nanometer geometries, since this often deviates from uid properties inlarger geometries.

The uids that are typically used in experiments are much more complicated thanthe simple atomic uids used in the MD simulations in this thesis. Although simulationmethods for more complicated materials exist, there are multiple reasons to simulatesimple atomic uids. In the rst place, the potential that describes the interactionbetween two atoms is well-understood and using this interaction potential to simulatesimple uids results in correct transport properties and phase transitions for a bulkuid. This is often not the case for more complicated materials. Due to the simplicityof monoatomic uids, microscopic origins of ow phenomena can be identied easierthan in a uid in which many more internal mechanisms, types of interactions and

possibly chemical reactions would be active. The insights obtained from simple systemscan then be used to increase the understanding of more complicated materials andgeometries. Furthermore, bonds between atoms tend to vibrate at high frequencies,such that the simulation time steps need to be smaller than for monoatomic uids.Fixing the bond lengths and angles, or even further coarse graining of the molecules,can allow for a larger simulation time step and a larger accessible number of molecules,but goes at the expense of the realism of the computer simulations. For example, manymodels exist that represent water molecules as a combination of atoms and electricalcharges with a xed internal structure [57]. Although these models avoid the small

5 Notable exceptions are ows through or around carbon nanotubes and porous silica.

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1.6. OUTLINE OF THE THESIS

simulation time steps needed to capture the fast bond vibrations, they often onlypartially succeed to recreate the bulk transport properties of water over a range of densities, temperatures and pressures. Furthermore, these water models are optimizedto reproduce bulk properties, but tend to be less suitable to reproduce interactionbetween water and silica or clay.

Besides the fact that simulation systems are often much simplied in comparison toreal systems, the accessible shear rates in computer simulations and experiments areoften also very different [ 17]. A method to overcome this gap is discussed in Chapter 6.

1.6 Outline of the thesis

Chapters 2, 3 and 4A variety of tools are introduced in these three chapters, that are needed for the simula-tions and analyses presented in later chapters. An introduction to molecular dynamicssimulations is given in Chapter 2. The chapter treats, amongst other things, some of the common and relevant interaction potentials, integrators and temperature controlmechanisms. Next, in Chapter 3, equations of motion and boundary conditions arepresented that can be used to simulate homogeneous simple shear ow, planar elonga-tional ow and combinations of shear and planar elongational ow. An introduction

to statistical mechanics methods is given in Chapter 4. The methods discussed in thischapter are used for the calculation of structural and dynamical uid properties froma sufficiently large set of simulation data.

Chapter 5

This chapter focusses on the ow behavior of simple atomic bulk uids in equilib-rium and under constant homogeneous planar ows. In particular, deviations fromNewtonian behavior are studied and quantied:

The density-dependence of the stress relaxation function of a simple uid. Thestress relaxation function as well as equilibrium material constants are calculatedfrom the equilibrium stress autocorrelation function.

A model is presented that predicts the pressure tensor for a non-Newtonianbulk uid under a homogeneous ow eld. The model provides a quantitativedescription of the strain thinning viscosity, bulk dilatancy, deviatoric viscoelasticlagging and out-of-shear-plane pressure anisotropy.

The transient shear stress and normal stress differences in a sheared bulk uid arestudied, both for startup and for cessation of ows. Non-equilibrium molecular11


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dynamics (NEMD) simulation results of the shear stress are compared with alinear viscoelastic prediction from equilibrium molecular dynamics (EMD).

Chapter 6

The transient-time correlation function (TTCF) method can be used to calculate thetransient and steady-state values of various quantities for a uid subjected to a con-stant deformation rate. This method is more efficient than direct averages of NEMDsimulations when the deformation rate is sufficiently small. In this chapter, TTCF isused to calculate the nonlinear response of homogeneous uids. Three ow problemsare discussed:

The TTCF response of components of the pressure tensor are studied, for asimple atomic uid subjected to a constant planar mixed ow (PMF) of shearand elongation. The TTCF response is compared to directly averaged NEMDmeasurements.

The normal stress differences in a sheared atomic uid are calculated usingTTCF. A phase space mapping is introduced in order to improve the statisticsof these computationally expensive calculations.

We study how the transients (the startup from equilibrium to non-equilibriumsteady state) of the pressure tensor and the viscosity of uids consisting of shortlinear chain molecules depend on the deformation rate, on the type of ow andon the length of the molecules. The modes of relaxation present in the stressautocorrelation function of a diatomic liquid are analyzed in order to increaseunderstanding of the transient viscosity.

Chapter 7

In this chapter, the local properties of a strongly conned uid are studied. The dis-tribution of atoms is strongly inhomogeneous near an interface, such that the uiddensity is a function of the location in the system. This, in turn, affects the local val-ues of other state variables, and the relations between them. The chapter provides anoverview of the relevant literature and required techniques for the study of an inhomo-geneous uid. In particular, we study the ow of a dense Lennard-Jones uid connedin a rectangular channel of approximately four nanometer width. Macroscopic eldsare obtained from microscopic data by temporal and spatial averaging and smoothingthe data with a self-consistent coarse-graining method based on kernel interpolation.

Two phenomena make the system interesting: (i) strongly conned uids show layer-ing, i.e., strong oscillations in density near the walls, and (ii) the stress deviates from

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the Newtonian uid assumption, not only in the layered regime, but also much furtheraway from the walls. Various scalar, vectorial, and tensorial elds are analyzed andrelated to each other in order to understand better the effects of both the inhomoge-neous density and the anisotropy on the ow behavior and rheology. The eigenvaluesand eigendirections of the stress tensor are used to quantify the anisotropy in stressand form the basis of a newly proposed objective, inherently anisotropic constitutivemodel that allows for non-collinear stress and strain-rate tensor by construction.

Chapter 8

In this nal chapter, a summary is given of the work that was presented and themain conclusions that were drawn from the observations. Furthermore, we list someimportant questions that remain open and recommend which steps need to be takento address these open problems.

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2

Molecular DynamicsSimulations

With the rise of micro- and nanotechnological applications, the industrial demand andscientic interest in understanding the microscopic origins of various uid phenomenahas seen a vast and rapid increase over the last decades. While experiments arevaluable in gaining understanding of the rheological behavior of uids, they are oftennot suitable to study what happens on a microscopic level. Simulations can lead to

insights that can sometimes not be extracted from experimental measurements.Continuum methods use macroscopic conservation equations in conjunction with

constitutive relations to study the behavior of a uid. The validity of these consti-tutive relations relies on the assumption that the uid properties are approximatelyconstant across macroscopically small volume elements. This assumption is accuratefor most practical purposes, but is invalid if variations in macroscopic quantities arelarge over atomic time or length scales. Furthermore, constitutive relations often pro-vide a heavily simplied model based on empirical ndings that are often limited tocertain conditions. Finally, to solve the closed set of governing continuum equations,transport coefficients need to be provided. In a non-Newtonian uid, these transportcoefficients are not only dependent on the thermodynamic state point of the uid, butalso on the ow eld.

The limitations of a continuum approach are avoided by using a microscopic ap-proach, such as molecular dynamics (MD). Methods based on a microscopic approachdeal with discrete objects, rather than innitesimally small volume elements. By inte-grating the equations of motion in time from a well-dened initial state, the methodkeeps track of particle positions and velocities. Macroscopic quantities, like pressure

and temperature, can be calculated from this microscopic information. Furthermore,these methods give rise to dynamical, structural and chemical information that is not

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(a) Solid-Vapor mixture (b) Liquid

Figure 2.1: Different phases, both simulated with the same interaction potential anddensity, only the temperature differs.

available in continuum methods.Different phases of a material can be created simply by changing the thermody-

namic conditions, such as the temperature or density, without having to change thesimulation methodology. Figure 2.1 shows two snapshots of a simulation of a dilute

uid, where the temperature is changed.Classical Newtonian dynamics forms the basis of MD. Newtons second law of

motion states that the motion of any object satises

F = ma (2.1)

= mr , (2.2)

where F is the total force acting on the object, a is its acceleration and m its mass.This equation can be written as a set of rst-order differential equations

r = v , (2.3)

v = F

m , (2.4)

where v is the velocity of the object. If the force on the object depends only onits position, the system is said to be holonomic. Furthermore, if the forces are fullydetermined by the state of the system, the set of equations is called deterministic (asopposed to stochastic). This means that one can integrate the state of a system forwardin time and then back to end up with the same initial state. Thus, a deterministic

system of equations in conjunction with a set of initial conditions ( r (0) , v (0)) has aunique solution for every moment in time.

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Rather than considering a single object, MD typically deals with many-body sys-tems, in which the dynamics of the bodies are coupled to each other via interactions.The equations of motion then become a large set of coupled rst-order differentialequations. The internal energy in a multi-body system is described by a Hamiltonian

H0 =N

i=1

p i p i2m i

+ U (r 1 , . . . , r N ) , (2.5)

where p i = mi c i = mi (v i u (r i )) denotes the peculiar (or thermal) momentumof particle i relative to the streaming motion u (r i ) and U is the potential energydue to interactions between particles. The streaming velocity is dened as u (r ) =

N i=1 m i v i (r r i )/ N i=1 m i (r r i ), where (r r i ) is the Dirac-delta function and

v i the laboratory velocity of particle i. The Hamiltonian equals the total energy inthe system if the uid is in equilibrium because the streaming motion is then zero andthus the peculiar momentum equals the laboratory momentum.

For a uid in equilibrium, equations of motion are derived from the internal energyHamiltonian (Eq. ( 2.5))

r i = H0

p i=

p im i

, (2.6)

p i = H0 r i

= F i , (2.7)

where F i is the resultant force on particle i due to all other bodies. By default, thenumber of particles and the system volume are constant in time. Furthermore, theseequations of motion are time-reversible and conserve energy. The latter can be shownby calculating the derivative of the Hamiltonian with respect to time

dH0dt

=N

i=1

H0 r i r i +

H0 p i p i =

N

i =1

H0 r i

H0 p i

H0 p i

H0 r i

= 0 . (2.8)

This property shows that each set of motion equations that can be derived from a

Hamiltonian conserves energy. Additionally, the equations of motion derived from aHamiltonian satisfy the following identity

r i r i

+ p i p i

= 2H0 r i p i

2H0 p i r i

= 0 . (2.9)

Solving the equations of motion analytically for all positions and momenta is gen-erally not feasible. Therefore, numerical schemes are needed to integrate the equationsof motion in time. In order to integrate the equations of motion, one needs to knowthe forces that act on each particle. The eld of MD simulations has developed rapidly

over the past ve decades. This has accelerated the development of interaction po-tentials, integration schemes and thermostats. Some of the relevant techniques are

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2.1. INTEGRATION SCHEMES

discussed in this chapter, but for a more detailed treatment, the reader is referred toone of the many excellent textbooks related to the subject [ 5861].

The outline of this chapter is as follows: The most common integration schemesare presented in Section 2.1. In Section 2.2, the most common atomic interaction po-tentials are discussed. The units used in MD simulations are discussed in Section 2.3.The calculation of the pressure tensor in molecular dynamics simulations is discussedin Section 2.4. Next, in Section 2.5, some of the challenges and implications of temper-ature control in MD simulations are discussed. Standard periodic boundary conditionsfor equilibrium molecular dynamics (EMD) simulations are introduced in Section 2.6.Finally, in Section 2.7, the concept of non-equilibrium molecular dynamics (NEMD) isintroduced.

2.1 Integration schemes

In MD simulations, the governing equations are solved numerically by integrating par-ticle positions and velocities in time. Since MD simulations typically require manytime steps, it is important that an integration scheme conserves quantities, like energyand momentum. Furthermore, time-reversible integration is required for theoreticaltreatment of a deterministic set of equations. Finally, a large time step is preferable,

without too much loss of accuracy. The error of an integration algorithm is a combi-nation of the order b of the algorithm and the step size t, so the global error is then

O(( t)b). Since we are often interested in averages rather than individual trajectories,a large step size is often preferred over a high accuracy.

Some integrators are said to be symplectic for Hamiltonian systems. This meansthat they preserve the Hamiltonian, regardless of the time step. This property ispractical (but not strictly required), especially given the large number of time steps inMD simulations. However, we will later see that many systems that we deal with arenon-Hamiltonian (see Sections 2.5.2 and 3.1).

The appropriate simulation time step t that can be used to integrate the equationsof motion depends on several factors. The simulation time step has to be chosensuch that the fastest microscopic processes can be calculated with a good temporalresolution. Furthermore, the time step has to be such that the integrator remainsstable. We do not engage in a detailed study of the time step, as this is well-establishedfor simple atomic uids. Time steps of approximately t = 0 .001 0.005 arecommonly used, depending on the integrator, the required accuracy and the detailsof the simulation, where is the unit time (see Section (2.3)). The integration time

step for molecular uids may be smaller. If bond lengths and angles are exible, theyare often responsible for the fastest modes in the system. If they are constrained, the

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2.1. INTEGRATION SCHEMES

maximum time step may depend on the constraint solver.The various simulations presented in this thesis have been performed with different

integrators. The reason for this is the fact that these integrators have different prop-erties and are not all equally suitable for different types of simulations. An overviewof the integrators and properties of the algorithms are given in this section.

2.1.1 (Velocity) Verlet

A well-known integration scheme in molecular dynamics simulations is the Verlet [62]scheme

r (t + t) = 2 r (t)

r (t

t) + a (t)( t)2 . (2.10)

This scheme is derived from a Taylor series expansion of r around t and has a dis-cretization (truncation) error of O(( t)4). The Verlet scheme follows from subtractingthe expansions for r (t t) from that for r (t + t). Since the terms that have an oddpower of t cancel out, the accuracy of this scheme is an order higher than a simpleTaylor series expansion up to the second time derivative of r . The Verlet scheme doesnot include the integration of velocity, which is often calculated from the positionsusing a nite difference scheme. A more commonly used method, based on the Verletscheme, is Velocity Verlet

r i (t + t) = r i (t) + v i (t) t + 12

a i (t)( t)2 , (2.11)

v i (t + t) = v i (t) + a i (t) + a i (t + t)

2 t . (2.12)

The Velocity Verlet scheme has a discretization error of O(( t)3) for the velocity, asopposed to an discretization error of O(( t)2) for the standard Verlet scheme with acentral difference calculation for the velocity.

2.1.2 Runge-Kutta

Higher-order methods are sometimes desirable for enhanced accuracy. Such schemescould be constructed by including more terms in the Taylor series expansion, but thiswould require the calculation of higher derivatives of the force, which is computa-tionally expensive. Alternatively, a single-step method can be devised that matchesthe accuracy of the higher-order Taylor series expansion by sequentially evaluatingthe function of interest g at several points within the time increment t, instead

of computing higher-order derivatives. Methods of this type are called Runge-Kuttamethods. A large variety of such schemes exists. We only present the fourth-order

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2.2. NON-BONDED INTERACTIONS BETWEEN ATOMS

explicit Runge-Kutta scheme

A (t + t) = A (t) + 16 (k 1 + 2 k 2 + 2 k 3 + k 4) ,

k 1 = t g(t, A (t)) ,k 2 = t g(t + t/ 2, A (t) + k 1 / 2) ,k 3 = t g(t + t/ 2, A (t) + k 2 / 2) ,k 4 = t g(t + t, A (t) + k 3) ,

(2.13)

where A can represent, for example, positions r or velocities v of particles, and g is theright-hand side of the governing rst-order differential equation (e.g., equation of mo-tion). Note that each increment in time requires four function evaluations. This makesthe fourth-order scheme signicantly more computationally expensive than lower-orderschemes.

This scheme is called explicit because each coefficient k i depends on previouslycalculated coefficients and on function evaluations from the previous step ( r (t), v (t)).Due to this feature, the method is easy to implement. However, the explicit Runge-Kutta scheme is only conditionally stable. Implicit Runge-Kutta schemes, on theother hand, are more difficult to implement, but they are more stable and much moreaccurate than explicit schemes.

In this thesis, the explicit fourth-order Runge-Kutta scheme is applied to the sim-ulations related to Chapter 6 because it is accurate as well as a single-step algorithm.

Single-step algorithms calculate the next information ( A (t + t)) based on the present(A (t)), without needing prior information ( A (t t)). This feature makes the algo-rithm self-starting, meaning that no additional algorithm is needed to start the inte-gration. An accurate self-starting algorithm is required for the study of startup-ow(see Section 5.4.1).

2.2 Non-bonded interactions between atoms

In this section, we treat the interactions between atoms in systems where the nochemical bonds are formed, i.e., monoatomic gases, liquids and amorphous solids. Non-bonded interactions are typically weaker than bonded interactions, such as covalent orionic bonds, and the number of interactions between any atom and its neighbors mayvary between atoms and varies in time. We focus on simple monoatomic uids. Theatoms in these uids are spherically symmetric, neutrally charged and do not exhibitchemical processes. Some examples of such uids are Argon, Xenon and Krypton. Theinteractions between the atoms are described by an energy potential. Several potentialsare known that can produce certain transport coefficients or phase transitions that are

in good agreement with empirical ndings. Thus, which potential to use depends onthe system and on the quantities or phenomena of interest.

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The potential energy can be a function of the position of individual particles, andthe relative position of particle pairs, triplets and even larger groups of simultaneouslyinteracting particles

U =N

i =1

U (r i ) +N

i =1

N

j =1j = i

U (r i , r j ) +N

i =1

N

j =1j = i

N

k=1k= i,j

U (r i , r j , r k ) + . . . . (2.14)

A suitable pair-interaction potential is known to be able to predict the properties of asimple uid very accurately [63]. Lee and Cummings [64] compared the shear viscositycalculated with a pair-interaction potential and a three-body interaction potential.They found that the three-body potential resulted only in a slightly lower viscosity

over the range of shear rates reported. Furthermore, Marcelli et al. [6567] havestudied extensively the inuence of three-body interactions on various quantities. Theyfound a relation, independent of shear rate, between transport properties calculatedwith two-body and three-body interaction potentials. A correction could be applied tothe energy, pressure and shear viscosity calculated with a two-body potential, ratherthan performing computationally expensive simulations with a three-body interactionpotential.

We will only consider pair-potentials in this study. The value of these potentialsare a function solely of the absolute distance between an interacting pair of atoms.

The most common pair potential for simple uids is the Lennard-Jones (LJ) potential[68]

U LJ = 4r

12

r

6 , (2.15)

where r = |r ij | = |r i r j | is the absolute distance between atoms i and j , isthe well-depth of the potential and the atomic length scale, which is chosen as thedistance at which the function value is zero. This potential is strongly repulsive at shortdistances ( r < 21/ 6) and attractive at longer distances ( r > 21/ 6). The attractivepart represents the Van der Waals forces between atoms. This term corresponds tothe 6 th power in the potential, which is based on empirical ndings. The repulsiveinteractions arise from Coulombic repulsions and, indirectly, from Pauli repulsion andthe exclusion of electrons from regions of space where the orbitals of closed-shell atomsoverlap. The repulsion corresponds to the 12 th power in the potential, which is chosensuch that the power is related to that of the attractive term, which is convenient froma computational viewpoint.

The powers of the potential can also be chosen differently. Some powers that havebeen used in the literature are (12,6), (9,6), and (28,7), or more complicated versions,

such as the n 6 LJ potential [69]. However, the 12-6 LJ potential is by far the mostcommonly used potential for MD simulations of simple uids. Fincham and Heyes [70]21


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have compared the shear viscosity of experimental liquid Argon to that calculated fromsimulations of a LJ uid. The authors found good agreement. Despite the simplicityof the LJ potential, different phases can be formed depending on the state point of theuid (see also Figure 2.1). It has been shown that the phase diagram 1 of a LJ uid isin good agreement with experimental results [63, 71, 72].

The LJ potential is often truncated in order to reduce computation time. Toprevent a discontinuity at the location where the potential is truncated, the wholepotential is shifted down by the value of the potential at the point of truncationU LJ (r c), such that the truncated and shifted Lennard-Jones (LJTS) potential is givenby

U LJTS =4 r

12

r6

r c12

+ r c6

for r r c0 for r > r c ,

(2.16)

where the cut-off distance is often chosen in the range rc 2.5 . . . 5. Truncatingand shifting the Lennard-Jones potential can inuence the transport properties of theuid and its phase diagram [7375]. Similarly, changing the repulsive power of thepotential affect its transport properties and phase diagram [ 69, 76].

A special case of the truncated and shifted Lennard-Jones potential has been in-troduced by Weeks, Chandler and Anderson (WCA) [77]. They have truncated thepotential at the distance of the LJ potential energy minimum rc = 2 1/ 6 and shifted

the remaining part to maintain a continuous potential energy function. By truncat-ing at the deepest point of the LJ potential, the attractive part of the interaction iseliminated, leaving a purely repulsive potential, given by

U WCA = 4 r

12

r6 + for r/ 21/ 6

0 for r/ > 21/ 6 .(2.17)

The LJ, LJTS (with rc = 2 .5) and WCA potential are shown in Figure 2.2.A shorter cut-off distance results in fewer interactions, which consequently results

in a reduction of the computation time. Despite this obvious advantage of a shortcut-off distance, the purely repulsive WCA potential has a limitation relative to aLJ potential that is truncated at a longer distance. Hansen and Verlet [ 71] showedthat potentials with a repulsive and an attractive component are needed to reproduce arealistic phase diagram. Earlier attempts with purely repulsive potentials succeeded inpredicting the phase transitions and the single phases, but did not manage to predict

1 Since the number of atoms and the system volume are xed by default, the density is a controlledquantity. Furthermore, the temperature of the uid will be controlled in the simulations (see Sec-tion 2.5). When a phase diagram is mentioned in this thesis, this refers to a two-dimensional diagram

with density and temperature on the axes, as this is the most natural choice for the simulations inthis thesis.

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1 1.5 2 2.5 3

1

0

1

2

3

r /

U /

LJLJTSWCA

2.3 2.4 2.5 2.6 2.7

0.04

0.02

0

0.02

Figure 2.2: Three versions of the Lennard-Jones potential. The full LJ potential, thetruncated and shifted Lennard-Jones (LJTS) potential at rc = 2 .5 and the WCApotential, which is truncated and shifted at rc = 2 1/ 6, which corresponds to the

minimum of the LJ potential.

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1 1.5 2 2.5 3

2

1

0

1

2

3

4

5

r /

F

/

LJLJTS

WCA

2.3 2.4 2.5 2.6 2.70.1

0.08

0.06

0.04

0.02

0

0.02

Figure 2.3: The interaction forces calculated from the potentials in Figure 2.2.

the coexistence of two uid phases [78]. Travis and Gubbins [ 79] compared several

properties, like density, velocity and heat ux for a conned simple liquid simulatedwith a LJ and WCA potential. The authors found large differences in all propertiesfor channels of widths 4 and 5.1. This was especially true for the narrowest of thetwo channels, in which the number of layers in the density prole were found to bedependent on the interaction potential used.

The force exerted on an atom due to interaction with another atom follows directlyfrom the interaction potential as

F ij = dU dr

r ijr

, (2.18)

where F ij is the force acting on atom i due to atom j and r ij = r i r j is the contactvector. The scalar force F = dU/dr as a function of the distance between two atomsis shown in Figure 2.3. The force proles are identical for distances smaller than thecut-off distance. This is because the shift of the potential has no inuence on itsslope. Since the WCA potential is truncated at the minimum of the LJ function, the

corresponding force prole shows no discontinuity, as opposed to the force prole of the LJTS potential with any other cut-off distance than that of WCA.

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2.3. LENNARD-JONES UNITS

2.3 Lennard-Jones Units

MD simulations are often performed in reduced units. Since the characteristic scalesare often very small in the conventional SI units, reduced units are not only moreconvenient for the user, but also avoid working in the vicinity of the numerical precisionlimitations of the computer. Each quantity is reduced with combinations of the lengthscale , the energy scale and the atomic mass m. The parameters that correspondto Argon are given in Table 2.1. While these parameters are known to be reasonablyaccurate for the reproduction of transport coefficients and phase transitions throughoutthe phase diagram, they do not always lead to the best possible agreement with someexperimental measurements [71] and theoretical models [80]. Parameters for other

Table 2.1: LJ parameters for liquid Argon, taken from Ref. [ 81].Basic Units Symbol parameter for Argon

Length 3.405 10 10 mEnergy /kB 119.8 K

1.65 10 21 JMass m 6.69 10 26 kg

uids can be found, for example, in Refs. [63, 72, 82, 83].All dimensional quantities can be reduced to dimensionless quantities by means

of these standard LJ units. To reduce a quantity A, with dimension kg m s , onecan write A = Am + / 2 + / 2 , where the asterisk denotes a non-dimensionalizedquantity [ 84]. The most relevant reductions for this work are listed in Table 2.2. Theunreduced values of most quantities in Table 2.2 are very large or very small. Thesevalues are often inconvenient to work with and might in some cases even result incalculations with numbers of the same order as the machine precision. In simulations

of simple uids, one often only works with reduced quantities, which are chosen suchthat they are around the order of unity. They can then be converted back into realunits at the end of the simulation.

2.4 Pressure and stress tensors

The hydrostatic pressure of a system in equilibrium is thermodynamically dened as

p H V N,T , (2.19)

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2.4. PRESSURE AND STRESS TENSORS

Table 2.2: Reduced units of several quantities. The last column shows the physicalvalues that correspond to unity in reduced units.

Variable Reduced units Real unitsDensity = 3 /m 1678 kg/ m3

Temperature T = T kB / 119.8 K

Viscosity = 2 / m 9.076 10 4 poisePressure p = p3 / 41.9 MPa

Time t = t /(m 2) 2.14 10 12 sStrain rate =

m 2 / 4.66 1011 s 1Force f = f/ 4.9

10 16 N

where H is the Helmholtz free energy, 2 V the system volume, N the number of particlesand T the temperature of the uid, where the subscripts N and T represent xedquantities. The thermodynamic denition of pressure is only valid in equilibrium andis inconvenient to calculate from a MD simulation. The mechanical interpretation of pressure is more common in MD

p = lim A 0 F n

A =

dF ndA

, (2.20)

where n is unit vector normal to the surface and A the surface area.The hydrostatic pressure in a homogeneous uid in equilibrium can be calculated

with the virial equation

pV = N kB T + 13

N

i =1 j = i

r ij F ij , (2.21)

where the rst term on the right is the kinetic part, with k B Boltzmanns constant,and the second term the congurational part. Eq. (2.21) assumes isotropy of the uid

properties and thus is not valid for an inhomogeneous uid or a uid out of equilibrium.The mechanical interpretation of the pressure (Eq (2.20)) can easily be generalized

to a position-dependent tensorial quantity. A pressure tensor can be dened from aninnitesimal force dF acting across an innitesimal surface dA , at location r

dF (r ) dA P (r ) . (2.22)Similar to the virial equation, the pressure tensor can be split into a kinetic part

P K (r ) = k B T (r )I /m due to convectional momentum transport, where I is the identity2

The Helmholtz free energy represents the amount of energy that can be transferred into work bya thermodynamic process.

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tensor, and a congurational part P U (r ) associated with the interactions betweenparticles. The pressure tensor is given by P (r ) = P K (r ) + P U (r ). Due to the differentnature of both contributions, some extreme scenarios can be identied. In a dilute gas,the distances between atoms are generally much larger than in a liquid or solid. Hence,the number of interactions and the corresponding forces are relatively small and thecongurational part of the pressure tensor will be small in comparison to the kineticpart. The opposite applies in a highly compressed dense solid/liquid, at moderatetemperatures: the close packing results in large forces and thus a large congurationalpressure tensor, whereas the transport of momentum due to uctuations is relativelysmall. In a typical liquid, both terms are of the same order of magnitude and neitherpart can be neglected. Note, however, that the congurational pressure tensor depends

strongly on the interaction potential. For example, by truncating the attractive partof a Lennard-Jones potential, the diagonal components of the congurational pressuretensor will increase in value and might even change from a negative to a positive value,depending on the state point of the uid. This dependence on the potential is irrelevantfor satisfying the continuum conservation equations since only the divergence of thepressure tensor occurs in these expressions.

The divergence of the pressure tensor can be derived from the evolution of themomentum density J (r ) as

J (r )t = t

N

i 1m i v i (r r i )

=N

i 1

m i v i (r r i ) N

i 1

m i v i v i (r r i )

=N

i 1

F i (r r i ) N

i 1

m i v i v i (r r i ) (2.23)

= 1

2

N

i 1 j = i

F ij ( (r r i ) (r r j )) uu +N

i 1

p i p im i

(r r i )= (uu + P ) .

The fourth equality uses the fact that F i = j = i F ij , Newtons third law F ij = F jiand writes the velocity in terms of a uctuation part and a streaming part v i =p i /m i + u (r i ). Since the pair interaction forces are symmetric, the pressure tensor isby denition also symmetric. From the last equality, we can write

P (r ) = N

i 1

p i p im i

(r r i ) 12

N

i 1 j = i

F ij ( (r r i ) (r r j )) . (2.24)

This expression does not have a unique solution for the pressure tensor P (r ), since it

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only denes the divergence of the pressure tensor.In 1950, Irving and Kirkwood [85] pioneered an ingenious way to calculate the local

microscopic pressure tensor from Eq. ( 2.24) by using a Taylor series expansion.

P (r ) =N

i=1

p i p im i

(r r i ) + 12

j = i

r ij F ij Oij [r ] (r r i ) , (2.25)

where r ij = r i r j and the operator Oij follows from the integral over the followingidentity

(r r i ) (r r j ) = r ij r

(r r i ) + 12!

r ij r ij 2

r 2 (r r i ) + . . .

= r r ij 1 12!r ij r + . . . + 1n!(r ij r )n 1 + . . . (r r i )

= r r ij Oij [r ] (r r i ) . (2.26)

Their formulation is generally applicable for single-component atomic uids in whichthe interactions between particles are described by a pair potential.

If the uid is homogeneous (i.e., particles are homogeneously distributed over space),the Taylor series expansion reduces to Oij = 1 so that the pressure tensor is given by

P (r ) =N

i=1

p i p im i

(r r i ) + 12

j = i

r ij F ij (r r i ) , (2.27)

This result can be referred to as the IK1 pressure tensor [ 86]. Since the pressure tensorfor a homogeneous uid is constant across the volume, the tensor can be averaged overthe volume, resulting in

P = 1

V

N

i =1

p i p i

m i+

1

2 j = ir ij F ij . (2.28)

When the spatial distribution of particles is inhomogeneous (for example near asolid interface), the expansion of Oij is required to calculate the local pressure tensor.An accurate approximation of Oij requires a large number of expansion terms, whichare numerically expensive to calculate. Alternatively, (r r i ) (r r j ) can beexpressed in terms of an integral over the path between the positions of the particlesi and j [8790]

(r r i ) (r r j ) = r r ij 10 (r r i + r ij ) d . (2.29)28


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The microscopic pressure tensor for an inhomogeneous uid can be derived by substi-tuting this identity into Eq. ( 2.24), which gives

P (r ) =N

i =1

p i p im i

(r r i ) + 12

j = i

r ij F ij 1

0d (r r i + r ij ) . (2.30)

This formulation was rst introduced by Schoeld and Henderson [ 87], in 1982.Mistura [91] derived the same expression for the pressure tensor in an inhomoge-

neous uid in a different way. He argued that the path, over which the integral goes, isunambiguous as it comes only in the denition of the distance between two particles,which is uniquely dened. Harasima [92] developed another method that produces the

same normal stress as the IK method, whereas the tangential stresses are different.While the local pressure tensors computed with both methods are not identical, dueto a different distribution of the information, the surface tension calculated with bothmethods is the same. This means that the integral of the difference between normaland tangential stress across the channel is equal for both methods. Tsai [93] comparedthe virial equation (Eq. ( 2.21)) to the IK method. He found that the tensorial methodleads to more precise results when comparable efforts in computation are compared.Furthermore, the IK pressure tensor is more suitable out of equilibrium and for inho-mogeneous uids compared to the scalar virial equation (Eq. ( 2.21)). Todd et al. [86]

derived an algorithm, called the method of planes (MoP), that avoids the ambiguouschoice of the interaction path. Local pressure components are computed from theconsideration of interaction forces across a plane passing through the point of inter-est. This method, however, can only be used to calculate the shear stress and one of the diagonal components of the pressure tensor, rather than the full tensorial quantity.The authors compared, for a LJ uid conned in a narrow slit pore, components of thepressure tensor, calculated with the MoP to those calculated with the IK1 approxima-tion and to results based on the integration of the Navier-Stokes momentum balanceequation, which does not require any atomistic information. This method is referredto as the IMC method. Good agreement was found between the MoP and IMC meth-ods, while the IK1 approximation showed spurious oscillations. Another algorithm wasintroduced by Cormier et al. [94]. They derived a pressure tensor based on averagingthe local pressure tensor over a spherical volume. Recently, Heyes and coworkers [95]have shown, for the limiting case of innitesimally thin bins, the equivalence betweenthe MoP and the volume averaging (VA) method, introduced by Cormier et al. [94].

A similar comparison as presented by Todd et al. [86] is shown in Figure 2.4.We compare the IK1 approximation to the IK pressure for an inhomogeneous uid

(Eq.( 2.30)). Our simulations correspond to a LJ uid with density = 0 .8 andtemperature T = 1 .0 in reduced units. The uid is conned in the x-direction between

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two parallel atomistic walls, separated by a distance W = 11 .1 and the uid is drivenin the negative y-direction by a constant body force f = 0 .2.

5 4 3 2 1 0 1 2 3 4 52

1

0

1

2

x

P x y

IK 1IK ful l

(a)

5 4 3 2 1 0 1 2 3 4 50

1

2

3

4

5

6

x

P x x

IK 1IK ful l

(b)

Figure 2.4: Shear stress P xy (a) and normal stress P xx (b) across the channel.

In recent years, the different pressure tensors have been extensively discussed andcompared in the literature. For example, Sonne et al. [96] compared the IK andHarasima method for the calculation of the pressure tensor in a lipid bilayer in the

liquid crystalline phase. The authors found a qualitative agreement between bothmethods. When a uid is strongly layered, qualitative differences can arise betweenboth denitions of the local pressure, as shown by Varnik et al. [97]. They comparedthe pressure tensors for a polymer lm calculated with the IK, MoP and Harasimaexpressions. Hafskjold and Ikeshoji [98] compared the IK and the Harasima pressuretensors for a hard-sphere uid. They concluded that the expressions are equal inCartesian coordinates, but the Harasima method does not result in a correct pressuretensor in spherical coordinates.

The pressure tensor is dened as a compressive quantity, i.e., a positive pressuretensor is commonly associated with compressive (i.e., repulsive) forces. This meansthat positive diagonal terms contribute positively to the hydrostatic pressure.

The pressure tensor is (besides the sign convention in some literature) identicalto the stress tensor , which is more common in rheology and the solid mechanicsliterature [ 99, 100], whereas the pressure tensor is very common in MD. The literatureis divided between the two tensorial quantities. Positive stresses are often associatedwith tensile forces, such that the relation between the pressure tensor and the stresstensor is given by P . It must be noted that the stress tensor is sometimes alsodened as a compressive quantity (for example in the eld of geology), in which case itis identical to the pressure tensor. In this work, we only use the compressive denition

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2.5. THERMOSTATTING

for both quantities, such that = P .The non-equilibrium pressure can be calculated from the pressure tensor as

p = 13

tr( P ) . (2.31)

Note that this is not by denition equal to the equilibrium pressure p0 , which isdened as one third of the trace of the pressure tensor for a uid in equilibrium. Sincethe pressure tensor for a homogeneous uid in equilibrium is isotropic, we can write p0 = P xx = P yy = P zz . The difference between p and p0 due to deformation of theuid is related to shear dilatancy [101], and will be studied in Chapter 5.

2.5 ThermostattingIn many MD simulations, control of the uid temperature is required. For example,when energy is added to the uid, or when a chang

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